Joint estimation and correction of geometric distortions for EPI functional MRI using harmonic retrieval

Structured Low-Rank Algorithms: Theory, Magnetic Resonance Applications, and Links to Machine Learning

In this survey, we provide a detailed review of recent advances in the recovery of continuous domain multidimensional signals from their few non-uniform (multichannel) measurements using structured low-rank matrix completion formulation. This framework is centered on the fundamental duality between the compactness (e.g., sparsity) of the continuous signal and the rank of a structured matrix, whose entries are functions of the signal. This property enables the reformulation of the signal recovery as a low-rank structured matrix completion, which comes with performance guarantees. We will also review fast algorithms that are comparable in complexity to current compressed sensing methods, which enables the application of the framework to large-scale magnetic resonance (MR) recovery problems. The remarkable flexibility of the formulation can be used to exploit signal properties that are difficult to capture by current sparse and low-rank optimization strategies. We demonstrate the utility of the framework in a wide range of MR imaging (MRI) applications, including highly accelerated imaging, calibration-free acquisition, MR artifact correction, and ungated dynamic MRI.

Calibration-Free B0 Correction of EPI Data Using Structured Low Rank Matrix Recovery

We introduce a structured low rank algorithm for the calibration-free compensation of field inhomogeneity artifacts in echo planar imaging (EPI) MRI data. We acquire the data using two EPI readouts that differ in echo-time. Using time segmentation, we reformulate the field inhomogeneity compensation problem as the recovery of an image time series from highly undersampled Fourier measurements. The temporal profile at each pixel is modeled as a single exponential, which is exploited to fill in the missing entries. We show that the exponential behavior at each pixel, along with the spatial smoothness of the exponential parameters, can be exploited to derive a 3-D annihilation relation in the Fourier domain. This relation translates to a low rank property on a structured multi-fold Toeplitz matrix, whose entries correspond to the measured k-space samples. We introduce a fast two-step algorithm for the completion of the Toeplitz matrix from the available samples. In the first step, we estimate the null space vectors of the Toeplitz matrix using only its fully sampled rows. The null space is then used to estimate the signal subspace, which facilitates the efficient recovery of the time series of images. We finally demonstrate the proposed approach on spherical MR phantom data and human data and show that the artifacts are significantly reduced.

MAGPI: A framework for maximum likelihood MR phase imaging using multiple receive coils

PURPOSE

Combining MR phase images from multiple receive coils is a challenging problem, complicated by ambiguities introduced by phase wrapping, noise, and the unknown phase-offset between the coils. Various techniques have been proposed to mitigate the effect of these ambiguities but most of the existing methods require additional reference scans and/or use ad hoc post-processing techniques that do not guarantee any optimality.

THEORY AND METHODS

Here, the phase estimation problem is formulated rigorously using a maximum-likelihood (ML) approach. The proposed framework jointly designs the acquisition-processing chain: the optimized pulse sequence is a single multiecho gradient echo scan and the corresponding postprocessing algorithm is a voxel-per-voxel ML estimator of the underlying tissue phase.

RESULTS

Our proposed framework (Maximum AmbiGuity distance for Phase Imaging, MAGPI) achieves substantial improvements in the phase estimate, resulting in phase signal-to-noise ratio (SNR) gains by up to an order of magnitude compared to existing methods.

CONCLUSION

The advantages of MAGPI are: (1) ML-optimal combination of phase data from multiple receive coils, without a reference scan; (2) voxel-per-voxel ML-optimal estimation of the underlying tissue phase, without the need for phase unwrapping or image smoothing; and (3) robust dynamic estimation of channel-dependent phase-offsets.

High-resolution, large dynamic range field map estimation

PURPOSE

We present a theory and a corresponding method to compute high-resolution field maps over a large dynamic range.

THEORY AND METHODS

We derive a closed-form expression for the error in the field map value when computed from two echoes. We formulate an optimization problem to choose three echo times which result in a pair of maximally distinct error distributions. We use standard field mapping sequences at the prescribed echo times. We then design a corresponding estimation algorithm which takes advantage of the optimized echo times to disambiguate the field offset value.

RESULTS

We validate our method using high-resolution images of a phantom at 7T. The resulting field maps demonstrate robust mapping over both a large dynamic range, and in low SNR regions. We also present high-resolution offset maps in vivo using both, GRE and multiecho gradient echo sequences. Even though the proposed echo time spacings are larger than the well known phase aliasing cutoff, the resulting field maps exhibit a large dynamic range without the use of phase unwrapping or spatial regularization techniques.

CONCLUSION

We demonstrate a novel three-echo field map estimation method which overcomes the traditional noise-dynamic range trade-off.

A joint acquisition-reconstruction paradigm for correcting inhomogeneity artifacts in MR echo planar imaging

One of the main sources of signal degradation in rapid MR acquisitions, such as Echo Planar Imaging (EPI), is magnetic field variations caused by field inhomogeneities and susceptibility gradients. If unaccounted for during the reconstruction process, this spatially-varying field can cause severe image artifacts. In this paper, we show that correcting for the resulting degradations can be formulated as a blind image deconvolution problem. We propose a novel joint acquisition-processing paradigm to solve this problem. We describe a practical implementation of this paradigm using a multi-image acquisition strategy and a corresponding joint estimation-reconstruction algorithm. The estimation step computes the spatial distribution of the field maps, while the reconstruction step yields a Minimum Mean Squared Error (MMSE) estimate of the imaged slice. Our simulations show that this proposed joint acquisition-reconstruction method is robust and efficient, offering factors of improvement in the quality of the reconstructed image as compared to other traditional methods.